Chapter 6.5, Problem 266E

In the following exercises, suppose a solid object in ${ℝ}^3$ has a temperature distribution given by $T(x,y,z)$ . The heat flow vector field in the object is $\text{F}=-k\nabla T$ , where $k>0$ is a property of the material. The heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is $\nabla \cdot \text{F}=-k\nabla \cdot \nabla T=-k{\nabla }^{2}T$ .

266. Compute the heat flow vector field.